Reductions of Galois representations and the theta operator

Abstract

Let [Formula: see text] be a prime, and let [Formula: see text] be a cuspidal eigenform of weight at least [Formula: see text] and level coprime to [Formula: see text] of finite slope [Formula: see text]. Let [Formula: see text] denote the mod [Formula: see text] Galois representation associated with [Formula: see text] and [Formula: see text] the mod [Formula: see text] cyclotomic character. Under an assumption on the weight of [Formula: see text], we prove that there exists a cuspidal eigenform [Formula: see text] of weight at least [Formula: see text] and level coprime to [Formula: see text] of slope [Formula: see text] such that [Formula: see text] up to semisimplification. The proof uses Hida–Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions of the local Galois representations associated to cusp forms with slopes in the interval [Formula: see text] were determined by Deligne, Buzzard and Gee and for slopes in [Formula: see text] by Bhattacharya, Ganguli, Ghate, Rai and Rozensztajn. We show that these reductions, in spite of their somewhat complicated behavior, are compatible with the displayed equation above. Moreover, the displayed equation above allows us to predict the shape of the reductions of a class of Galois representations attached to eigenforms of slope larger than [Formula: see text]. Finally, the methods of this paper allow us to obtain upper bounds on the radii of certain Coleman families.